This is the monograph on the theory and applications of copula entropy (CE). This book first introduces the theory of CE, including its background, definition, theorems, properties, and estimation methods. The theoretical applications of CE to structure learning, association discovery, variable selection, causal discovery, system identification, time lag estimation, domain adaptation, multivariate normality test, copula hypothesis test, two-sample test, change point detection, and symmetry test are reviewed. The relationships between the theoretical applications and their connections to correlation and causality are discussed. The framework based on CE for measuring statistical independence and conditional independence is compared to the other similar ones. The advantages of CE based methodologies over the other comparable ones are evaluated with simulations. The mathematical generalizations of CE are reviewed. The real applications of CE to every branch of science and engineering are briefly introduced.